Lenders tighten or loosen their standards for issuing credit as economic conditi

Question

Lenders tighten or loosen their standards for issuing credit as economic conditions change. One of the criteria lenders use to evaluate the creditworthiness of a potential borrower is her credit risk score: in the U.S. it is usually a FICO score. FICO scores range from 300 to 850. A consumer with a high FICO score is perceived to be a low credit risk to the lender and is more likely to be extended credit than a consumer with a low score. A credit card represents a line of credit, because the credit card holder obtains a loan whenever the card is used to pay for a purchase. A study of credit card accounts opened in 2002 found a mean FICO score for credit card holders (at the time the card was issued) of 731 and a standard deviation of 76. [Source: Sumit Agarwal, John C. Driscoll, Xavier Gabaix, and David Laibson, "Learning in the Credit Card Market, " Working Paper 13822, National Bureau of Economic Research (NBER), February 2008.] You conduct a hypothesis test to determine whether banks have tightened their standards for issuing credit cards since 2002. You collect a random sample of 64 credit cards issued during the past 6 months. The sample mean FICO score of the credit card holders (at the time their cards were issued) is x bar = 746. Assume that the standard deviation of the population of FICO scores for credit cards issued during the past 6 months is known to be sigma = 76, the standard deviation from the NBER study. Let mu equal the true population mean FICO score for consumers issued credit cards in the past 6 months. You should formulate the null and alternative hypotheses as: H_0: mu = 731, H_1: mu > 731 H_0: mu = 731, H_1: mu 731, H_1: mu = 731 H_0: x bar = 731, H_1: x bar > 731 If the null hypothesis is true, the sampling distribution of X bar is approximated by ____________________ distribution with _______________ and a standard deviation of ______________. The value of the standardised test statistic is_____________.

Explanation / Answer

Solution:-

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: = 731
Alternative hypothesis: < 731

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is small.

Formulate an analysis plan. For this analysis, the significance level is 0.10. The test method is a one-sample t-test.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t-statistic test statistic (t).

SE = s / sqrt(n) = 76 / sqrt(64) = 9.5
DF = n - 1 = 64 - 1 = 63
t = (x - ) / SE = (746 - 731)/9.5 = 1.57895

where s is the standard deviation of the sample, x is the sample mean, is the hypothesized population mean, and n is the sample size.

Here is the logic of the analysis: Given the alternative hypothesis ( < 731), we want to know whether the observed sample mean is small enough to cause us to reject the null hypothesis.

The observed sample mean produced a t statistic test statistic of 1.57895. We use the t Distribution Calculator to find P(t < 1.57894) = 0.059684.

Interpret results. Since the P-value (0.059684) is less than the significance level (0.10), we cannot accept the null hypothesis.

Conclusion. Reject null hypothesis. We do conclude that banks have tightened their standards for issuing credit cards since 2002.

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